Extreme points of convex sets
Extreme points of convex sets
复制标题
凸集的极值点
DOI：
10.1007/bf01362699
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发表时间：
1973
影响因子：
1.4
通讯作者：
W. Pranger
中科院分区：
文献类型：
--
作者：
W. Pranger
介绍。凸集是这样的集合：如果它包含两个点，它包含连接这两个点的段[1，p. 2，和2]。方括号内的数字为末尾的参考书目。Minkowski定义了凸集的某些点，他称之为极值点[1，pp. 15-16]；[3]，第157页。它们与其他一些点有关这些点在距离意义上被称为极值点以区别于前者，后者在闵可夫斯基意义上被称为极值点。对抽象赋范线性空间中凸集的这两类极值点进行了详细的研究。首先，需要根据球面邻域的凹凸性(1)区分两类赋范线性空间。一个赋范的线性空间，使得连接球面邻域的任意两点的线段在邻域内，除了给定的点本身以外，称为空间L*。所有其他赋范线性空间都归为一类并用L表示。在空间L*中对极值点的研究要比在空间L中简单得多，结果也更完备。在第10节中考虑的一个例子表明，作为空间L*的性质可能只取决于距离函数的性质，而不取决于空间的线性性质。在2中考虑了极值点的存在性。由闵可夫斯基首次证明的欧几里得3空间的近似定理推广到3中的L*和L空间。根据两类极值点之间的关系，在4中区分了两类凸集，并证明了Minkowski意义上的极值点集可以是封闭的，也可以是不封闭的。在5中，考虑了给定集合的闭凸包，推广了Minkowski近似定理(3)。在6中建立了紧集的一个一般定理。证明了在完备度量空间中，如果一个集合可以用闭紧集一致地逼近它，那么这个集合就是紧的。这个定理和Minkowski近似定理(5)使我们在7中证明了Banach空间中紧集的闭凸包是紧的。第8节简要讨论了闵可夫斯基近似定理的意义。在第9节中给出了一系列定理，这些定理更精确地建立了凸集与其极值点之间的关系。第10章考虑了一些例子。这篇论文可以看作是对抽象空间几何的研究。
Introduction. A convex set is a set such that if it contains two points, it contains the segment joining these points [1, p. 2, and 2. Numbers in square brackets refer to the bibliography at the end]. Minkowski defined certain points of convex sets which he called extreme points [1, pp. 15-16; 3, p. 157]. They are related to certain other points which are here called extreme points in the sense of distance to distinguish them from the former, which are called extreme points in the sense of Minkowski. A detailed study is made of these two types of extreme points of convex sets in abstract normed linear spaces. In the first place, it is necessary to distinguish two types of normed linear spaces on the basis of the convexity properties of spherical neighborhoods (1). A normed linear space such that the segment joining any two points of a spherical neighborhood is interior to the neighborhood except at most for the given points themselves is called a space L*. All other normed linear spaces are classed together and denoted by L. The study of extreme points is far simpler in spaces L* than in spaces L, and the results are more complete. An example considered in 10 shows that the property of being a space L* may depend on the properties of the distance function alone and not on the linearity properties of the space. In 2 the existence of extreme points is considered. An approximation theorem first proved by Minkowski for euclidean 3-space is extended to spaces L* and L in 3. Two kinds of convex sets are distinguished in 4 on the basis of the relation of the two kinds of extreme points, and it is shown that the set of extreme points in the sense of Minkowski may be either closed or not closed. In 5 the closed convex hull of a given set is considered, and Minkowski’s Approximation Theorem (3) is extended. A general theorem on compact sets is established in 6. It is shown that in a complete metric space a set is compact if it is possible to approximate uniformly to it by means of closed compact sets. This theorem and Minkowski’s Approximation Theorem (5) enable us to show in 7 that the closed convex hull of a compact set in a Banach space is compact. The significance of Minkowski’s Approximation Theorem is considered briefly in 8. A series of theorems is given in 9 which establish more precisely the relation between a convex set and its extreme points. Some examples are considered in 10. The pper may be considered a study in the geometry of abstract space.